Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.85·3-s − 3.75·5-s − 3.33·7-s + 0.442·9-s + 2.31·11-s − 1.30·13-s + 6.97·15-s − 17-s − 1.32·19-s + 6.18·21-s − 2.50·23-s + 9.11·25-s + 4.74·27-s + 6.82·29-s + 7.79·31-s − 4.29·33-s + 12.5·35-s + 2.02·37-s + 2.41·39-s + 6.45·41-s − 4.87·43-s − 1.66·45-s − 7.41·47-s + 4.11·49-s + 1.85·51-s − 0.619·53-s − 8.70·55-s + ⋯
L(s)  = 1  − 1.07·3-s − 1.68·5-s − 1.25·7-s + 0.147·9-s + 0.698·11-s − 0.361·13-s + 1.80·15-s − 0.242·17-s − 0.303·19-s + 1.34·21-s − 0.522·23-s + 1.82·25-s + 0.913·27-s + 1.26·29-s + 1.39·31-s − 0.747·33-s + 2.11·35-s + 0.333·37-s + 0.386·39-s + 1.00·41-s − 0.743·43-s − 0.247·45-s − 1.08·47-s + 0.587·49-s + 0.259·51-s − 0.0851·53-s − 1.17·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 + 1.85T + 3T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
7 \( 1 + 3.33T + 7T^{2} \)
11 \( 1 - 2.31T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + 2.50T + 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 - 7.79T + 31T^{2} \)
37 \( 1 - 2.02T + 37T^{2} \)
41 \( 1 - 6.45T + 41T^{2} \)
43 \( 1 + 4.87T + 43T^{2} \)
47 \( 1 + 7.41T + 47T^{2} \)
53 \( 1 + 0.619T + 53T^{2} \)
61 \( 1 + 1.42T + 61T^{2} \)
67 \( 1 - 0.724T + 67T^{2} \)
71 \( 1 - 8.70T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 1.49T + 83T^{2} \)
89 \( 1 + 6.21T + 89T^{2} \)
97 \( 1 + 4.07T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.098967204245706258452212136519, −7.17128783733169244854039355075, −6.48645510473880959957755068491, −6.17023473403800097771804323413, −4.90870586650856285750853312905, −4.34386943371512007402842031737, −3.51756599128142099077613142950, −2.74291983693506004095876174417, −0.844065227341930856662468257343, 0, 0.844065227341930856662468257343, 2.74291983693506004095876174417, 3.51756599128142099077613142950, 4.34386943371512007402842031737, 4.90870586650856285750853312905, 6.17023473403800097771804323413, 6.48645510473880959957755068491, 7.17128783733169244854039355075, 8.098967204245706258452212136519

Graph of the $Z$-function along the critical line